Random Coefficient Demand Challenges - Estimation of Random...

Estimation of Random Coe¢ cient Demand Models:Challenges, Di¢ culties and WarningsChristopher R. Knittel and Konstantinos Metaxoglou°October 1, 2008AbstractEmpirical exercises in economics frequently involve estimation of highly nonlinear models.The criterion function may not be globally concave or convex and exhibit many local ex-trema. Choosing among these local extrema is non-trivial for a variety of reasons. In thispaper, we analyze the sensitivity of parameter estimates, and most importantly of eco-nomic variables of interest, to both starting values and the type of non-linear optimizationalgorithm employed. We focus on a class of demand models for di/erentiated productsthat have been used extensively in industrial organization, and more recently in publicand labor. We °nd that convergence may occur at a number of local extrema, at saddlesand in regions of the objective function where the °rst-order conditions are not satis°ed.We °nd own- and cross-price elasticities that di/er by a factor of over 100 depending onthe set of candidate parameter estimates. In an attempt to evaluate the welfare e/ectsof a change in an industry±s structure, we undertake a hypothetical merger exercise. Ourcalculations indicate consumer welfare e/ects can vary between positive values to negativeseventy billion dollars depending on the set of parameter estimates used.°Knittel:Department of Economics, University of California, Davis, CA and NBER.Email:crknit-[email protected]Metaxoglou:Bates White LLC.Email: [email protected]We havebene°tted greatly from conversations with Steve Berry, Severin Borenstein, Michael Greenstone, Phil Haile,Aviv Nevo, Hal White, Frank Wolak, Catherine Wolfram, and seminar participants at the University of Cal-gary, University of California at Berkeley, the University of California Energy Institute, and the 2008 NBERWinter IO meeting. Metaxoglou aknowledges °nancial support from Bates White, LLC. We are also gratefulto Bates White, LLC for making their computing resources available. All remaining errors are ours.

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11Introduction1.1What this paper is aboutEmpirical research in economics often requires estimating highly nonlinear models, where theobjective function may not be globally concave or convex. Obtaining parameter estimates inthese cases requires a nonlinear search algorithm along with sets of starting values and stoppingrules. For a common class of demand models used in industrial organization, and more recentlyin labor and public economics, we °nd that termination of many popular nonlinear search algo-rithms may occur at local extrema, saddles points, as well as in regions of the objective functionwhere the °rst-order conditions for optimality fail. Furthermore, parameter estimates and mea-sures of market performance, such as price elasticities, exhibit notable variation depending onthe combination of the algorithm and starting values in the optimization exercise at hand.

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